Here are some interactions between the study of the `uncountable' and core
mathematics.
1) Zilber's attempt to find an axiomatization for the complex
numbers with exponentiation (necessarily in infinitary
logic) has the interesting twist that the canonicity (his word
-categoricity in power is the precise interpretation) in aleph_1 is
rather
straightforward algebra. To extend to structures of cardinality the
continuum requires the notion of excellence - a notion of how to
amalgamate n-dimensional systems of models. And then the results extend to
arbitrary cardinality. This theory was first established by Shelah in a
more general situation but the connections with `core math' are due to
Zilber.
2) In particular, in `Model Theory, Geometry, and Arithmetic of the
universal cover of a semi-abelian variety' Zilber establishes an
equivalence between `categoricity in uncountable powers of a certain
infinitary sentences' and `arithmetic properties of semi-abelian
varieties. The notion of `arithmetic' here is that of `arithmetic
algebraic geometry' not various logical hierarchies.
3) See http://www.maths.ox.ac.uk/~zilber/
and http://www2.math.uic.edu/~jbaldwin/org/res.html
for more detailed accounts.
I have a couple of follow-ups on this that I should post in a few days.
John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
312-413-2149
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607
Assistant to the director
Jan Nekola: 312-413-3750